Algebra Examples

$3 x^{2} + 2 = 11 - x^{2}$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Move all the expressions to the left side of the equation.

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Step 1.1.1

Subtract $11$ from both sides of the equation.

$3 x^{2} + 2 - 11 = - x^{2}$

Step 1.1.2

Add $x^{2}$ to both sides of the equation.

$3 x^{2} + 2 - 11 + x^{2} = 0$

Step 1.2

Simplify $3 x^{2} + 2 - 11 + x^{2}$ .

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Step 1.2.1

Add $3 x^{2}$ and $x^{2}$ .

$4 x^{2} + 2 - 11 = 0$

Step 1.2.2

Subtract $11$ from $2$ .

$4 x^{2} - 9 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 4$ , $b = 0$ , and $c = - 9$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (4 \cdot - 9)}}{2 \cdot 4}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raising $0$ to any positive power yields $0$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 4 \cdot - 9}}{2 \cdot 4}$

Step 4.1.2

Multiply $- 4 \cdot 4 \cdot - 9$ .

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Step 4.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{0 \pm \sqrt{0 - 16 \cdot - 9}}{2 \cdot 4}$

Step 4.1.2.2

Multiply $- 16$ by $- 9$ .

$x = \frac{0 \pm \sqrt{0 + 144}}{2 \cdot 4}$

Step 4.1.3

Add $0$ and $144$ .

$x = \frac{0 \pm \sqrt{144}}{2 \cdot 4}$

Step 4.1.4

Rewrite $144$ as $12^{2}$ .

$x = \frac{0 \pm \sqrt{12^{2}}}{2 \cdot 4}$

Step 4.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{0 \pm 12}{2 \cdot 4}$

Step 4.2

Multiply $2$ by $4$ .

$x = \frac{0 \pm 12}{8}$

Step 4.3

Simplify $\frac{0 \pm 12}{8}$ .

$x = \frac{\pm 3}{2}$

Step 5

The final answer is the combination of both solutions.

$x = \frac{3}{2}, - \frac{3}{2}$